interest formulas and their applications .pdf

engineering economy

1. Chapter (4): Interest Formulas and their Applications

2. Introduction:
Interest rate is the rental value of money.
It represents the growth of capital per unit period.
The period may be a month, a quarter, semiannual
or a year. An interest rate 15% compounded
annually means that for every hundred rupees
invested now, an amount of Rs. 15 will be added
to the account at the end of the first year.

3. So, the total amount at the end of the first year
will be Rs. 115.
At the end of the second year, again 15% of Rs.
115, i.e. Rs. 17.25 will be added to the account.
Hence the total amount at the end of the
second year will be Rs. 132.25.
The process will continue thus till the specified
number of years.

4. Time value of money:
If an investor invests a sum of Rs. 100 in
a fixed deposit for five years with an a fixed deposit for five years with an
interest rate of 15% compounded annually,
the accumulated amount at the end of
every year will be as shown in Table .1.

5. Year endInterest (Rs.)Compound amount (Rs.)
0 0 100.00
1 15.00 115.00
2 17.25 132.00
Table .1 Compound Amounts: (amount of deposit = Rs. 100.00)
2 17.25 132.00
3 19.84 152.09
4 22.81 174.90
5 26.24 201.14

6. The formula to find the future worth in the
third column is:
F= P x
Where:Where:
P= principal amount invested at time 0,
F= future amount,
i= interest rate compounded annually,
n= period of deposit.

7. The maturity value at the end of the fifth
year is Rs. 201.14. This means that the
amount Rs. 201.14 at the end of the fifth
year is equivalent to Rs. 100.00
at time 0.
This is diagrammatically shown in Fig. 1.
This explanation assumes that the inflation is
at zero percentage.

8. 0 1 2 3 4 5
201.14
Fig. 1 Time value of money
100
i= 15%

9. Alternatively, the above concept may be
discussed as follows: If we want Rs.
100.00 at the end of the nth year, what is
the amount that we should deposit now the amount that we should deposit now
at a given interest rate, say 15%?
A detailed working is shown in Table .2.

10. End of year (n)Present worthCompound amount after n year(s)
0 100
1 86.96 100
2 75.61 100
3 65.75 100
Table .2 Present Worth Amounts: (rate of interest = 15%)
3 65.75 100
4 57.18 100
5 49.72 100
6 43.23 100
7 37.59 100
8 32.69 100
9 28.43 100
10 24.72 100

11. The formula to find the present worth in the second
column is:
From Table .2, it is clear that if we want Rs. 100
at the end of the fifthyear, we should now at the end of the fifthyear, we should now
deposit an amount of Rs. 49.72. Similarly, if we
want Rs. 100.00 at the end of the 10th year, we
should now deposit an amount of Rs. 24.72.

12. Also, this concept can be stated as follows:
A person has received a prize from a finance company during the
recent festival contest. But the prize will be given in either of
the following two modes:
1. Spot payment of Rs. 24.72 or
2. Rs. 100 after 10 years from now (this is based on 15% interest 2. Rs. 100 after 10 years from now (this is based on 15% interest
ratecompounded annually).
If the prize winner has no better choice that can yield more than
15%interest rate compounded annually, and if 15%
compounded annually is the common interest rate paid in all the
finance companies,

13. then it makes no difference whether he receives Rs.
24.72 now or Rs. 100 after 10 years.
On the other hand, let us assume that the prize winner
has his own business wherein he can get a yield of 24%
interest rate compounded annually, it is better for him
to receive the prize money of Rs. 24.72 at present and
utilize it in his business. If this option is followed, the
equivalent amount for Rs. 24.72 at the end of the 10th
year is Rs. 212.45. This example clearly demonstrates
the time value of money.

14. Interest formulas:
While making investment decisions, computations will
be done in many ways.
To simplify all these computations, it is extremely
important to know how to use interest formulas more important to know how to use interest formulas more
effectively. Before discussing the effective application of
the interest formulas for investment-decision making,
the various interest formulas are presented first.

15. Interest rate can be classified into simple interest rate
and compoundinterest rate.
In simple interest, the interest is calculated, based on
the initial deposit for every interest period. In this case,
calculation of interest on interest is not applicable. calculation of interest on interest is not applicable.
In compound interest, the interest for the current
period is computed based on the amount (principal
plus interest up to the end of the previous period) at
the beginning of the current period.

16. The notations which are used in various interest
formulae are asfollows:
P= principal amount.
n= No. of interest periods.
i = interest rate .
F= future amount at the end of year n.
A= equal amount deposited at the end of every
interest period.

17. 1. Single-Payment Compound Amount:
Here, the objective is to find the single
future sum (F) of the initial payment (P)
made at time 0 after n periods at an made at time 0 after n periods at an
interest rate i compounded every period.
The cash flow diagram of this situation is
shown in Fig. 2.

18. 01234 5
F
Fig. 2 Cash flow diagram of single-payment compound amount
. n.
P
i %

19. The formula to obtain the single-payment compound amount is:
= P (F/P , i ,n)
Where:
(F/P, i, n) is called as single-payment compound amount
factor.factor.
Example .1:
A person deposits a sum of Rs. 20,000 at the interest
rate of 18% compounded annually for 10 years. Find the
maturity value after 10 years.

20. Solution:
P= Rs. 20,000
i = 18% compounded annually
n = 10 years
= P(F/ P, i, n)= P(F/ P, i, n)
= 20,000 (F/P, 18%, 10)
= 20,000 x 5.234 = Rs. 1,04,680
The maturity value of Rs. 20,000 invested now at 18%
compounded yearly is equal to Rs. 1,04,680 after 10 years.

21. 2. Single-Payment Present Worth Amount:
Here, the objective is to find the present worth
amount (P) of a single future sum (F) which will
be received after n periods at an interest rate of be received after n periods at an interest rate of
n compoundedat the end of every interest
period.
The corresponding cash flow diagram is shown
in Fig. 3.

22. 01234
F
Fig. 3 Cash flow diagram of single-payment present worth amount
. n.
P
i %
n

23. The formula to obtain the present worth is
Where:
(P/F, i, n) is termed as single-payment present worth factor.
Example 2:
A person wishes to have a future sum of Rs. A person wishes to have a future sum of Rs.
1,00,000 for his son’s education after 10 years from
now. What is the single-payment that he should
deposit now so that he gets the desired amount
after 10 years? The bank gives 15% interest rate
compounded annually.

24. Solution: F= Rs. 1,00,000
i = 15%, compounded annually
n= 10 years
= 1,00,000 (P/F, 15%, 10)= 1,00,000x0.2472= 1,00,000 (P/F, 15%, 10)= 1,00,000x0.2472
= Rs.24,720
The person has to invest Rs. 24,720 now so that he will
get a sum ofRs. 1,00,000 after 10 years at 15% interest
rate compounded annually.

25. 3.Equal-Payment Series Compound Amount:
In this type of investment mode, the objective
is to find the future worth of n equal payments
which are made at the end of every interest
period till the end of the nth interest period at
an interest rate of i compounded at the end of
each interest period. The corresponding cash
flow diagram is shownin Fig. 4.

26. 0 1234
F
i%
Fig. 4 Cash flow diagram of equal-payment series compound amount
.
n
.
AAAA
P
A

27. In Fig. 4:
A= equal amount deposited at the end of each interest period
n = No. of interest periods
i = rate of interest
F= single future amount
The formula to get F is:The formula to get F is:
Where:
(F/A, i, n) is termed as equal-payment series compound
amount factor.

28. Example 3:
A person who is now 35 years old is planning for
his retiredlife. He plans to invest an equal sum of
Rs. 10,000 at the end of every year for the next
25 years starting from the end of the next year.
The bank gives 20% interest rate, compounded
annually. Find the maturity value of his account
when he is 60 years old.

29. Solution:
A= Rs. 10,000
n= 25 years
i = 20%i = 20%
F= ?
The corresponding cash flow diagram is
shown in Fig. 5.

30. 0 1234
F
i = 20%
Fig. 5 Cash flow diagram of equal-payment series compound amount
.
25
.
10,00010,00010,00010,000
10,000

31. F= = A(F/A, i, n)
= 10,000(F/A, 20%, 25)
= 10,000 x 471.981= 10,000 x 471.981
= Rs. 47,19,810
The future sum of the annual equal payments
after 25 years is equal toRs. 47,19,810.

32. 4. Equal-Payment Series Sinking Fund:
In this type of investment mode, the objective is
to find the equivalentamount (A) that should
be deposited at the end of every interest period
for n interest periods to realize a future sum (F) for n interest periods to realize a future sum (F)
at the end of the nth interest period at an
interest rate of i.
The corresponding cash flow diagram is shown
in Fig. 6.

33. 01234
F
Fig. 6 Cash flow diagram of equal-payment series sinking fund
.
n
.
AAAA
P
i% A

34. In Fig. 6,
A= equal amount to be deposited at the end of each interest
period
n= No. of interest periods
i = rate of interest
F= single future amount at the end of the nth period
The formula to get F is:
where
(A/F, i, n) is called as equal-payment series sinking fund factor.

35. Example .4:
A company has to replace a present facility after 15
years at an outlay of Rs. 5,00,000. It plans to deposit
an equal amount at the end of every year for the
next 15 years at an interest rate of 18% compounded
annually. Find the equivalent amount that must be
deposited at the end of every year for the next 15
years.

36. Solution:
F= Rs. 5,00,000
n= 15 years
i = 18%i = 18%
A= ?
The corresponding cash flow diagram is shown in
Fig. 7.

37. 0 1234
500,000
i = 18 %
Fig. 7 Cash flow diagram of equal-payment series sinking fund
.
15
.
AAAA
A

38. = 5,00,000(A/F, 18%, 15)
= 5,00,000 x 0.0164= 5,00,000 x 0.0164
= Rs. 8,200
The annual equal amount which must be
deposited for 15 years is Rs. 8,200.

39. Effective Interest Rate:
Let i be the nominal interest rate compounded
annually. But, in practice, the compounding may occur
less than a year. For example, compounding may be less than a year. For example, compounding may be
monthly, quarterly, or semi-annually. Compounding
monthly means that the interest is computed at the
end of every month. There are 12 interest periods in a
year if the interest is compounded monthly.

40. Under such situations, the formula to
compute the effective interest rate,
which is compounded annually, is
Effective interest rate, R= b+ i / c g –1
where,
i = the nominal interest rate
C= the number of interest periods in a
year.

41. Example 5:
A person invests a sum of Rs. 5,000 in a bank
at a nominal interest rate of 12% for 10 years.
The compounding is quarterly. Find the
maturity amount of the deposit after 10 years.
Solution:
P= Rs. 5,000
n= 10 years
i = 12% (Nominal interest rate)
F= ?

42. Method 1:
No. of interest periods per year = 4
No. of interest periods in 10 years = 10 x 4 = 40
Revised No. of periods (No. of quarters), N= 40Revised No. of periods (No. of quarters), N= 40
Interest rate per quarter, r= 12%/4
= 3%, compounded quarterly.
= Rs. 16,310.19

43. Method 2:
No. of interest periods per year, C= 4
Effective interest rate,
= 12.55%, compounded annually.= 12.55%, compounded annually.

44. Questions
1. Explain the time value of money.
2. Give practical applications of various interest
formulas.
3. A person deposits a sum of Rs. 1,00,000 in a bank for 3. A person deposits a sum of Rs. 1,00,000 in a bank for
his son’s education who will be admitted to a
professional course after 6 years. The bank pays 15%
interest rate, compounded annually.
Find the future amount of the deposited money at the
time of admitting his son in the professional course.

45. 4. A person needs a sum of Rs. 2,00,000 for his
daughter’s marriage which will take place 15 years from
now. Find the amount of money that he should deposit
now in a bank if the bank gives 18% interest,
compounded annually.compounded annually.
5. A person who is just 30 years old is planning for his
retired life. He plans to invest an equal sum of Rs.
10,000 at the end of every year for the next 30 years
starting from the end of next year.

46. The bank gives 15% interest rate, compounded
annually. Find the maturity value of his account when
he is 60 years old.
6. A company is planning to expand its business after 5
years from now. The money required for the expansion years from now. The money required for the expansion
programme is Rs. 4,00,00,000. What annual equivalent
amount should the company deposit at the end of
every year at an interest rate of 15% compounded
annually to get Rs. 4,00,00,000 after 5 years from now?

47. Thank You